Final answer:
James Perkins needs to calculate the annual savings required to achieve one million dollars in 15 years, considering his existing $200,000 IRA earning 8% annually. The calculations involve the compound interest formula for the existing investment and the future value of an annuity formula for the annual savings required. By combining these financial formulas, one can determine the amount to save each year to reach the retirement goal.
Step-by-step explanation:
The question aims to calculate the annual savings necessary for James Perkins to reach his goal of having one million dollars at retirement in 15 years, with an existing $200,000 in an IRA earning 8 percent annually. To solve this, we must find the future value of the current investment and determine the annual contribution that will fill the gap between this future value and the target amount, using the future value of an annuity formula.
Firstly, we calculate the future value of the current $200,000 investment using the compound interest formula:
FV = PV × (1 + r)n
Where, FV is the future value, PV is the present value ($200,000), r is the annual interest rate (0.08), and n is the number of years (15).
After 15 years, the $200,000 will grow to FV = $200,000 × (1 + 0.08)15.
Secondly, we determine the annual savings required to achieve the remaining amount to reach one million by using the future value of an annuity formula:
FV = PMT × × × × ((1 + r)n - 1) / r
In this formula, PMT is the annual payment, and the rest of the variables are as defined previously. Solving for PMT will give us the needed annual savings.
Upon calculation, the result will be rounded to the nearest dollar to find how much James needs to save each year.